Abstract

A comparatively simple way to generate time series of atmospherically distorted wave fronts is described and tested. Temporal power spectra of Zernike aberrations, extracted from the time series, agree very well with theoretical predictions. A method for generation of longer time series is also shown to give results in accordance with theory, except for the lowest temporal frequencies modeled. A way to superimpose several time series of wave fronts to generate a multilayer model of the atmosphere is briefly discussed.

© 1996 Optical Society of America

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  1. D. L. Fried, “Statistics of a geometric representation of wavefront distortion,” J. Opt. Soc. Am. 55, 1427–1435 (1965).
  2. B. L. McGlamery, “Computer simulation studies of compensation of turbulence degraded images,” in Image Processing, J. C. Urbach, ed., Proc. Soc. Photo-Opt. Instrum. Eng.74, 225–233 (1976).
  3. N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. 29, 1174–1180 (1990).
  4. J. Wallace, F. G. Gebhardt, “New method for numerical simulation of atmospheric turbulence,” in Modeling and Simulation of Optoelectronic Systems, J. D. O’Keefe, ed., Proc. Soc. Photo-Opt. Instrum. Eng.642, 261–268 (1986).
  5. B. J. Herman, L. A. Strugala, “Method for inclusion of low-frequency contributions in numerical representation of atmospheric turbulence,” in Propagation of High-Energy Laser Beams through the Earth's Atmosphere, P. B. Ulrich, L. E. Wilson, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1221, 183–192 (1990).
  6. C. A. Haniff, R. W. Wilson, “Closure-phase imaging with partial adaptive correction,” Publ. Astron. Soc. Pac. 106, 1003–1014 (1994).
  7. V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961).
  8. J.-M. Conan, P.-Y. Madec, G. Rousset, “Temporal power spectra of turbulent wavefronts,” in Proceedings of the Thirteenth National Solar Observatory Sacramento Peak Summer Workshop on Real Time and Post Facto Solar Image Correction, R. R. Radick, ed. (National Solar Observatory/Sacramento Peak, Sunspot, N.M., 1993), pp. 6–14.
  9. F. Roddier, M. J. Northcott, J. E. Graves, D. L. McKenna, D. Roddier, “One-dimensional spectra of turbulence-induced Zernike aberrations: time-delay and isoplanicity error in partial adaptive compensation,” J. Opt. Soc. Am. A 10, 957–965 (1993).
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  11. O. Engvold, T. Andersen, eds., “Status of the design of the Large Earth-Based Solar Telescope,” (Large Earth-Based Solar Telescope Foundation, Oslo, 1990).
  12. P.-Y. Madec, J.-M. Conan, G. Rousset, “Temporal characterization of atmospheric wavefront for adaptive optics,” in Proceedings of European Southern Observatory Conference 42 on Progress in Telescope and Instrumentation Technologies, M.-H. Ulrich, ed. (European Southern Observatory, Garching, Germany, 1992), pp. 471–474.
  13. H. Jakobsson, “Time series of atmospherically distorted wave-fronts,” in Adaptive Optics in Astronomy, F. Merkle, M. A. Ealey, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2201, 314–320 (1994).

1994 (1)

C. A. Haniff, R. W. Wilson, “Closure-phase imaging with partial adaptive correction,” Publ. Astron. Soc. Pac. 106, 1003–1014 (1994).

1993 (1)

1990 (1)

N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. 29, 1174–1180 (1990).

1976 (1)

1965 (1)

Conan, J.-M.

J.-M. Conan, P.-Y. Madec, G. Rousset, “Temporal power spectra of turbulent wavefronts,” in Proceedings of the Thirteenth National Solar Observatory Sacramento Peak Summer Workshop on Real Time and Post Facto Solar Image Correction, R. R. Radick, ed. (National Solar Observatory/Sacramento Peak, Sunspot, N.M., 1993), pp. 6–14.

P.-Y. Madec, J.-M. Conan, G. Rousset, “Temporal characterization of atmospheric wavefront for adaptive optics,” in Proceedings of European Southern Observatory Conference 42 on Progress in Telescope and Instrumentation Technologies, M.-H. Ulrich, ed. (European Southern Observatory, Garching, Germany, 1992), pp. 471–474.

Fried, D. L.

Gebhardt, F. G.

J. Wallace, F. G. Gebhardt, “New method for numerical simulation of atmospheric turbulence,” in Modeling and Simulation of Optoelectronic Systems, J. D. O’Keefe, ed., Proc. Soc. Photo-Opt. Instrum. Eng.642, 261–268 (1986).

Graves, J. E.

Haniff, C. A.

C. A. Haniff, R. W. Wilson, “Closure-phase imaging with partial adaptive correction,” Publ. Astron. Soc. Pac. 106, 1003–1014 (1994).

Herman, B. J.

B. J. Herman, L. A. Strugala, “Method for inclusion of low-frequency contributions in numerical representation of atmospheric turbulence,” in Propagation of High-Energy Laser Beams through the Earth's Atmosphere, P. B. Ulrich, L. E. Wilson, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1221, 183–192 (1990).

Jakobsson, H.

H. Jakobsson, “Time series of atmospherically distorted wave-fronts,” in Adaptive Optics in Astronomy, F. Merkle, M. A. Ealey, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2201, 314–320 (1994).

Madec, P.-Y.

P.-Y. Madec, J.-M. Conan, G. Rousset, “Temporal characterization of atmospheric wavefront for adaptive optics,” in Proceedings of European Southern Observatory Conference 42 on Progress in Telescope and Instrumentation Technologies, M.-H. Ulrich, ed. (European Southern Observatory, Garching, Germany, 1992), pp. 471–474.

J.-M. Conan, P.-Y. Madec, G. Rousset, “Temporal power spectra of turbulent wavefronts,” in Proceedings of the Thirteenth National Solar Observatory Sacramento Peak Summer Workshop on Real Time and Post Facto Solar Image Correction, R. R. Radick, ed. (National Solar Observatory/Sacramento Peak, Sunspot, N.M., 1993), pp. 6–14.

McGlamery, B. L.

B. L. McGlamery, “Computer simulation studies of compensation of turbulence degraded images,” in Image Processing, J. C. Urbach, ed., Proc. Soc. Photo-Opt. Instrum. Eng.74, 225–233 (1976).

McKenna, D. L.

Noll, R. J.

Northcott, M. J.

Roddier, D.

Roddier, F.

Roddier, N.

N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. 29, 1174–1180 (1990).

Rousset, G.

J.-M. Conan, P.-Y. Madec, G. Rousset, “Temporal power spectra of turbulent wavefronts,” in Proceedings of the Thirteenth National Solar Observatory Sacramento Peak Summer Workshop on Real Time and Post Facto Solar Image Correction, R. R. Radick, ed. (National Solar Observatory/Sacramento Peak, Sunspot, N.M., 1993), pp. 6–14.

P.-Y. Madec, J.-M. Conan, G. Rousset, “Temporal characterization of atmospheric wavefront for adaptive optics,” in Proceedings of European Southern Observatory Conference 42 on Progress in Telescope and Instrumentation Technologies, M.-H. Ulrich, ed. (European Southern Observatory, Garching, Germany, 1992), pp. 471–474.

Strugala, L. A.

B. J. Herman, L. A. Strugala, “Method for inclusion of low-frequency contributions in numerical representation of atmospheric turbulence,” in Propagation of High-Energy Laser Beams through the Earth's Atmosphere, P. B. Ulrich, L. E. Wilson, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1221, 183–192 (1990).

Tatarskii, V. I.

V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961).

Wallace, J.

J. Wallace, F. G. Gebhardt, “New method for numerical simulation of atmospheric turbulence,” in Modeling and Simulation of Optoelectronic Systems, J. D. O’Keefe, ed., Proc. Soc. Photo-Opt. Instrum. Eng.642, 261–268 (1986).

Wilson, R. W.

C. A. Haniff, R. W. Wilson, “Closure-phase imaging with partial adaptive correction,” Publ. Astron. Soc. Pac. 106, 1003–1014 (1994).

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (1)

Opt. Eng. (1)

N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. 29, 1174–1180 (1990).

Publ. Astron. Soc. Pac. (1)

C. A. Haniff, R. W. Wilson, “Closure-phase imaging with partial adaptive correction,” Publ. Astron. Soc. Pac. 106, 1003–1014 (1994).

Other (8)

V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961).

J.-M. Conan, P.-Y. Madec, G. Rousset, “Temporal power spectra of turbulent wavefronts,” in Proceedings of the Thirteenth National Solar Observatory Sacramento Peak Summer Workshop on Real Time and Post Facto Solar Image Correction, R. R. Radick, ed. (National Solar Observatory/Sacramento Peak, Sunspot, N.M., 1993), pp. 6–14.

J. Wallace, F. G. Gebhardt, “New method for numerical simulation of atmospheric turbulence,” in Modeling and Simulation of Optoelectronic Systems, J. D. O’Keefe, ed., Proc. Soc. Photo-Opt. Instrum. Eng.642, 261–268 (1986).

B. J. Herman, L. A. Strugala, “Method for inclusion of low-frequency contributions in numerical representation of atmospheric turbulence,” in Propagation of High-Energy Laser Beams through the Earth's Atmosphere, P. B. Ulrich, L. E. Wilson, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1221, 183–192 (1990).

B. L. McGlamery, “Computer simulation studies of compensation of turbulence degraded images,” in Image Processing, J. C. Urbach, ed., Proc. Soc. Photo-Opt. Instrum. Eng.74, 225–233 (1976).

O. Engvold, T. Andersen, eds., “Status of the design of the Large Earth-Based Solar Telescope,” (Large Earth-Based Solar Telescope Foundation, Oslo, 1990).

P.-Y. Madec, J.-M. Conan, G. Rousset, “Temporal characterization of atmospheric wavefront for adaptive optics,” in Proceedings of European Southern Observatory Conference 42 on Progress in Telescope and Instrumentation Technologies, M.-H. Ulrich, ed. (European Southern Observatory, Garching, Germany, 1992), pp. 471–474.

H. Jakobsson, “Time series of atmospherically distorted wave-fronts,” in Adaptive Optics in Astronomy, F. Merkle, M. A. Ealey, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2201, 314–320 (1994).

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Figures (7)

Fig. 1
Fig. 1

Mean phase-structure function for the individual wave fronts of a simulated time series (solid curve) compared with the theoretical phase-structure function for Kolmogorov turbulence (dashed line).

Fig. 2
Fig. 2

Mean power spectrum for ten simulated time series of atmospherically distorted wave fronts (solid curves) compared with theoretical predictions (dashed curves). Zernike polynomials 2, x tilt (upper curves), and 3, y tilt (lower curves, vertically shifted), are shown.

Fig. 3
Fig. 3

Same as Fig. 2, but for Zernike polynomials 4, defocus (upper curves), and 11, spherical aberration (lower curves, vertically shifted).

Fig. 4
Fig. 4

Same as Fig. 2, but for Zernike polynomials 7 (upper curves) and 8 (lower curves, vertically shifted), coma.

Fig. 5
Fig. 5

Power spectrum for a longer simulated time series of atmospherically distorted wave fronts (solid curves) compared with theoretical predictions (dashed curves). Zernike polynomials 2, x tilt (upper curves), and 3, y tilt (lower curves, vertically shifted), are shown.

Fig. 6
Fig. 6

Same as Fig. 5, but for Zernike polynomials 4, defocus (upper curves), and 11, spherical aberration (lower curves, vertically shifted).

Fig. 7
Fig. 7

Same as Fig. 5, but for Zernike polynomials 7 (upper curves) and 8 (lower curves, vertically shifted), coma.

Equations (7)

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D ϕ ( r ) = [ ϕ ( r + r ) ϕ ( r ) ] 2 ,
D ϕ = 6.88 ( | r | r 0 ) 5 / 3 ,
r 0 = 0.185 ( λ 2 h h + Δ h C n 2 ( z ) d z ) 3 / 5 .
ν c = 0.3 ( n + 1 ) υ / D ,
a k = Z k ( r ) ϕ ( R r ) w d 2 r ,
ϕ = k c k ϕ k ,
k c k 2 = 1.

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